We recall the definition of Bochner-Sobolev space \(H^s(0, T;L^2(\Omega ))\) , see for example [1]} for the settings.
For any \(0<s<1\) , one can define Sobolev-Slobodeckii seminorm \(|\cdot |_{H^{s}(0,T;L^2(\Omega ))}\)  by
\( |v|_{H^{s}(0,T;L^2(\Omega ))}=\int _0^T\int _0^T\frac{\Vert (v(t)-v(\tau ))\Vert ^2}{|t-\tau |^{1+2s}} \,{\mathrm {d}}\tau {\mathrm {d}}t,\) 
